Structures · Algebra

DivisionSemiring

A DivisionSemiring is a Semiring with multiplicative inverses for nonzero elements. An instance of DivisionSemiring K includes maps nnratCast : ℚ≥0 → K and nnqsmul : ℚ≥0 → K → K. Those two fields are needed to implement the DivisionSemiring K → Algebra ℚ≥0 K instance since we need to control the specific definitions for some special cases of K (in particular K = ℚ≥0 itself). See also note [forgetful inheritance]. If the division semiring has positive characteristic p, our division by zero convention forces nnratCast (1 / p) = 1 / 0 = 0.

Defined in

Mathlib.Algebra.Field.Defs

Shape

One type argument · adds div_eq_mul_inv, zpow_zero', zpow_succ', zpow_neg', inv_zero, mul_inv_cancel, nnratCast_def, nnqsmul, nnqsmul_def